Shearlet approximations to the inverse of a family of linear operators

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R E S E A R C H

Open Access

Shearlet approximations to the inverse of a

family of linear operators

Lin Hu

1,2

_{and Youming Liu}

1*

*_{Correspondence:}

liuym@bjut.edu.cn

1_{Department of Applied}

Mathematics, Beijing University of Technology, Beijing, 100124, P.R. China

Abstract

The Radon transform plays an important role in applied mathematics. It is a

fundamental problem to reconstruct images from noisy observations of Radon data. Compared with traditional methods, Colona, Easley andetc.apply shearlets to deal with the inverse problem of the Radon transform and receive more eﬀective reconstruction. This paper extends their work to a class of linear operators, which contains Radon, Bessel and Riesz fractional integration transforms as special examples. MSC: 42C15; 42C40

Keywords: inverse problems; shearlets; approximation; Radon transform; noise

1 Introduction and preliminary

The Radon transform is an important tool in medical imaging. Althoughf ∈L₍_R_{) can}

be recovered analytically from the Radon dataRf(θ,t), the solution is unstable and those data are corrupted by some noise in practice []. In order to recover the objectf stably and control the ampliﬁcation of noise in the reconstruction, many methods of regulariza-tion were introduced including the Fourier method, singular value decomposiregulariza-tion,etc.[]. However, those methods produced a blurred version of the original one.

Curvelets and shearlets were then proposed, which proved to be eﬃcient in dealing with edges [–]. In , Candés and Donoho applied curvelets [] to the inverse problem

Y=Rf+εW, (.)

where the recovered functionf is compactly supported and twice continuously diﬀeren-tiable away from a smooth edge;W denotes a Wiener sheet;εis a noisy level. Because curvelets have complicated structure, Colonna, Easley,etc.used shearlets to deal with the problem (.) in  and received an eﬀective reconstructive algorithm [].

Note that the Bessel transform and the Riesz fractional integration transform arise in many scientiﬁc areas ranging from physical chemistry to extragalactic astronomy. Then this paper considers a more general problem,

Y=Kf+εW, (.)

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whereKstands for a linear operator mapping the Hilbert spaceL_(R_{) to another Hilbert}

spaceY and satisﬁes

K*Kf∧(ξ) =b+|ξ|–αfˆ(ξ) (.) withb> ,α>  (K*_{is the conjugate operator of}_K_{). Here and in what follows,}_fˆ_denotes

the Fourier transform off. The next section shows that Radon, Bessel and Riesz fractional integration transforms satisfy the condition (.).

The current paper is organized as follows. Section  presents three examples for (.) and several lemmas. An approximation result is proved in the last section, which contains Theorem . of [] as a special case.

At the end of this section, we introduce some basic knowledge of shearlets, which will be used in our discussions. The Fourier transform of a functionf ∈L_(R₎_∩_L_(R_{) is deﬁned}

by

ˆ f(ξ) =

Rf(x)e

–πix·ξ dx.

The classical method extends that deﬁnition toL₍_R_{) functions.}

There exist many diﬀerent constructions for discrete shearlets. We introduce the con-struction [] by taking two functionsψ,ψof one variable such thatψˆ,ψˆ∈C∞(R) with

their supportssuppψˆ⊂[–_, –_]∪[_,_],suppψˆ⊂[–, ] and

j≥

ψˆ

–jω= 

|ω| ≥



,

j–

l=–j ψˆ

jω–l=  |ω| ≤.

Here,C∞(Rn_{) stands for inﬁnitely many times diﬀerentiable functions on the Euclidean}

spaceRn_{. Then two shearlet functions}_ψ()_,_ψ()_{are deﬁned by} ˆ

ψ()(ξ) :=ψˆ(ξ)ψˆ

ξ

ξ

and ψˆ()(ξ) :=ψˆ(ξ)ψˆ

ξ

ξ

respectively.

To introduce discrete shearlets, we need two shear matrices

B=

   

, B=

   

and two dilation matrices

A=

   

, A=

   

.

Deﬁne discrete shearletsψ_j(_,d_l_,)_k(x) := j_ψ(d)₍_Bl

dA j

dx–k) forj≥, –j≤l≤j– ,k∈Z

andd= , . Then there existsϕˆ∈C_∞(R_{) such that}

ϕj,k(x),ψ

(d)

j,l,k(x),j≥j≥, –j≤l≤j– ,k∈Z,d= , 

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forms a Parseval frame ofL_(R_{), where}_ϕ

j,k(x) := jϕ(jx–k). More precisely, forf ∈

L₍_R_),

f(x) =

k∈Z

f,ϕj,kϕj,k(x) +



d=

j≥j

j–

l=–j

k∈Z

f,ψ_j(_,d_l_,)_kψ_j(_,d_l_,)_k(x)

holds inL₍_R_{). It should be pointed out that}_ψ(d)

j,l,k(x) are modiﬁed forl= –jand j– , as

seen in [].

2 Examples and lemmas

In this section, we provide three important examples of a linear operator K satisfying (K*_Kf₎∧₍_ξ_{) = (}_b₊_|_ξ_|₎–α_fˆ₍_ξ_{) and present some lemmas which will be used in the next}

section. To introduce the ﬁrst one, deﬁne a subspace ofL_(R_),

DR:=f ∈LR,f is bounded⊆

f ∈LR,

R|

ξ|–fˆ(ξ)dξ< +∞

and a Hilbert space

L[,π)×R:=

f(θ,t), π



R

f(θ,t)dt dθ< +∞

with the inner productf,g:=_π_Rf(θ,t)g(θ,t)dt dθ.

Example . LetLθ,t:={(x,y),xcosθ+ysinθ=t} ⊆Randds(x,y) be the Euclidean

mea-sure on the lineLθ,t. Then the classical Radon transform R:D(R)→L([,π)×R) deﬁned

by

Rf(θ,t) =

Lθ,t

f(x,y)ds(x,y)

satisﬁes (R*Rf)∧(ξ) =|ξ|–fˆ(ξ).

Proof By the deﬁnition ofD(R),_R|ξ|–fˆ(ξ)gˆ(ξ)dξ < +∞forf,g∈D(R). It is easy to see that_π dθ_+∞fˆ(ωcosθ,ωsinθ)gˆ(ωcosθ,ωsinθ)dω=_π dθ_Rfˆ(ωcosθ,ωsinθ)×

ˆ

g(ωcosθ,ωsinθ)dω. This with the Fourier slice theorem ([, ]) and the Plancherel for-mula leads to

R|

ξ|–fˆ(ξ)gˆ(ξ)dξ= π

 dθ

+∞

 ˆ

f(ωcosθ,ωsinθ)gˆ(ωcosθ,ωsinθ)dω

= π

 dθ

R(Rθf)

∧₍_ω₎₍_R

θg)∧(ω)dω

= π

 dθ

RRf(θ,t)Rg(θ,t)dt=Rf,Rg,

whereRθf(t) :=Rf(θ,t). Moreover,(R*Rf)∧,gˆ=R*Rf,g=Rf,Rg=|ξ|–fˆ(ξ),gˆ(ξ)for

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Because D(R_{) is dense in}_L_(R_{), one receives the desired conclusion (}_R*_Rf₎∧₍_ξ_{) =} |ξ|–_fˆ₍_ξ_{). Here,}_R*_Rf _∈_L_(R_{) for}_f _∈_D_(R_{). In fact,}_R*_Rf_{= }_π_I

f by [], whereIis the

Riesz fractional integration transform deﬁned by

I(f)(x) :=Cα

R

f(x–y)

|y| dy

with some normalizing constant Cα []. We rewrite I(f)(x) =

|y|≤r

f(x–y)

|y| dy +

|y|>r f(x–y)

|y| dy=:J+J. Leth(y) =



|y|B(,)(y),hr(y) =



rh(

y

r), whereB(, ) stands for the unit

ball ofR_{and }

Arepresents an indicator function on the setA. ThenJ=

|y|≤r

f(x–y)

|y| dy=

r_|_y_|≤_rhr(y)f(x–y)dy≤rMf(x) by Theorem  of reference [, p.], whereMf is the

Hardy-Littlewood maximal function off.

On the other hand, the Holder inequality implies

J≤ f_p–p

|y|>r



|y|pdy

 p

≤r–pf p p–

withp> . Taker= [Mf(x)]–p– _{, one gets}_I

(f)(x)≤[Mf(x)]

p–

p–_{( +}f p

p–) andI(f)≤

( +f p p–)Mf

p– p– (p–)

p–

( +f p p–)f

p– p– (p–)

p–

< +∞, sincef ∈L

p

p–(R₎∩_L (p–)

p– (R_{) due to}

the assumptionf∈D(R_{) and}(p–)

p– > ,

p p–> .

In order to introduce the next example, we usef ∗g to denote the convolution off

andg.

Example . The Bessel operator Bα:L(R)→L(R) deﬁned byBαf =bα ∗f with ˆ

bα(ξ) = ( +|ξ|)– α

 and_α>  satisﬁes

B*_αBαf

∧

(ξ) = +|ξ|–αfˆ(ξ).

Proof It is known thatbα(x)∈L(R) forα>  []. Hence, (Bαf)∧(ξ) =bˆα(ξ)fˆ(ξ) = ( + |ξ|)–α__fˆ₍_ξ_{). For}_f_,_g_∈_L₍_R_),₍_B*

αBαf)∧,gˆ=B*αBαf,g=Bαf,Bαg=(Bαf)∧, (Bαg)∧= ( +|ξ|)–α_fˆ₍_ξ_),_g_ˆ₍_ξ₎_{. Thus,}

B*αBαf

∧

(ξ) = +|ξ|–αfˆ(ξ).

To introduce the Riesz fractional integration transform, we deﬁne

D=f∈LR,f has compact support⊆LR∩LR.

Then D⊆Ls₍_R_{) (}_≤_s_≤_{). For}_f _∈_D_{and  <}_α_{< , the Riesz fractional integration}

transform is deﬁned by

Iα(f)(x) :=Cα

R

f(y)

|x–y|–αdy∈L

_R_, _(.)

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Lemma . Let S(R₎_{be the Schwartz space and}₌_{_ψ_∈_S_(R_), ∂β

∂xβψ() = ,β∈Z+×

Z+_} _with_Z+_{being the non-negative integer set}_._Deﬁne _:=_{_ϕ₌_ψˆ_,_ψ _∈_}_._{Then with}

α> ,

(Iαf)∧(ξ) =|ξ|–αfˆ(ξ)

holds for each f ∈.

Example . The transformIαdeﬁned by (.) satisﬁes (Iα*Iαf)∧(ξ) =|ξ|–αfˆ(ξ) forf ∈D

and  <α<_.

Proof As proved in Examples ., ., it is suﬃcient to show that forf ∈D,

(Iαf)∧(ξ) =|ξ|–αfˆ(ξ). (.)

One proves (.) ﬁrstly forf∈C_∞(R). Takeμ(r)∈C∞([,∞)) with ≤μ(r)≤ and

μ(r) = ⎧ ⎨ ⎩

, r≥;

, ≤r≤.

Deﬁne ψN(ξ) :=μ(N|ξ|)fˆ(ξ). Then ψN(ξ)∈ and fN(x) :=ψˇN(x) =ψˆN(–x)∈ . By

Lemma .,

(IαfN)∧(ξ) =|ξ|–αfˆN(ξ). (.)

Letk(x) be the inverse Fourier transform of the function  –μ(|x|) andkN(x) :=_Nk(_Nx). Then _Rk(x)dx=  andfN(x) =f(x) –kN∗f(x). Moreover, the classical approximation

theorem [] tells

lim

N→∞fN–fp= 

forp> . On the other hand,IαfN–Iαf=Iα(fN–f)≤CfN–f 

+α due to Theorem 

[, p.]. Hence,limN→∞(IαfN)∧(ξ) – (Iαf)∧(ξ)=limN→∞IαfN–Iαf= . That is,

lim

N→∞(IαfN)

∧₍_ξ_{) = (}_I

αf)∧(ξ) (.)

in L_(R_{) sense. Note that}_|_ξ_|–α_fˆ

N(ξ) –|ξ|–αfˆ(ξ)=

R|ξ|–α|ˆf(ξ)|[ –μ(N|ξ|)]dξ;

|ξ|–α|ˆf(ξ)|_∈_L₍_R_{) with  <}_α_<

andlimN→∞[ –μ(N|ξ|)] = . Then lim

N→∞

_|ξ|–αfˆN(ξ) –|ξ|–αfˆ(ξ)_=  (.)

thanks to the Lebesgue dominated convergence theorem, which meanslimN→∞|ξ|–α×

ˆ

fN(ξ) =|ξ|–αfˆ(ξ) inL(R) sense. This with (.), (.) shows (.) forf ∈C∞(R).

In order to show (.) forf ∈D, one can ﬁndg∈C∞ (R) such that

Rg(x)dx=  and

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f ∗gN ∈C∞ (R), the above proved fact says

Iα(f∗gN)

∧

(ξ) =|ξ|–α(f ∗gN)∧(ξ). (.)

The same arguments as (.) and (.) show thatlimN→∞(Iα(f ∗gN))∧(ξ) = (Iαf)∧(ξ) and limN→∞|ξ|–α(f ∗gN)∧(ξ) =|ξ|–αfˆ(ξ). Hence,

(Iαf)∧(ξ) =|ξ|–αfˆ(ξ).

This completes the proof of (.) forf ∈D.

Next, we prove a lemma which will be used in the next section. For convenience, here and in what follows, we deﬁneM=N∪MwithN=Z_,_M_:=_{_μ_{= (}_j_,_l_,_k_,_d_{) :}_j_≥_j

, –j≤ l≤j_{– ,}_k_∈_Z_,_d_{= , }_}_{. Then the shearlet system (introduced in Section ) can be}

rep-resented as{sμ:μ∈M}, wheresμ=ψμ=ψ_j(_,d_l_,)_kifμ∈M, andsμ=ϕμ=ϕj,kifμ∈N. Lemma . Let K satisfy(K*_Kf₎∧₍_ξ_{) = (}_b₊_|_ξ_|₎–α_fˆ₍_ξ₎_and_{_s

μ,μ∈M}be shearlets in-troduced in the ﬁrst section. Deﬁneσˆμ(ξ) = (b+|ξ|)αˆsμ(ξ)and Uμ := –αjKσμ.Then Uμ ≤C and forμ∈M,

f,sμ= αjKf,Uμ.

Proof By the Plancherel formula and the assumptionσˆμ(ξ) = (b+|ξ|)αsˆμ(ξ), one knows

thatf,sμ=ˆf,ˆsμ=ˆf(ξ), (b+|ξ|)–ασˆμ(ξ). Moreover, f,sμ=

_ˆ

f(ξ),K*Kσμ

∧

(ξ)=f,K*Kσμ

=Kf,Kσμ= αjKf,Uμ

due to (K*Kf)∧(ξ) = (b+|ξ|)–α_fˆ₍_ξ_{) and}_U

μ:= –αjKσμ.

Next, one shows Uμ ≤ C. Note that Kσμ = Kσμ,Kσμ = K*Kσμ,σμ = (K*_K_σ

μ)∧,σˆμ, (K*Kσμ)∧= (b+|ξ|)–ασˆμ(ξ) andσˆμ(ξ) = (b+|ξ|)αˆsμ(ξ). ThenKσμ= ˆsμ(ξ), (b+|ξ|)αˆsμ(ξ)and

Uμ= –αjKσμ= –αj

R

b+|ξ|αˆsμ(ξ)

 dξ.

Becausesuppsˆμ⊆Cj:= [–j–, j–]\[–j–, j–], one receivesUμ= –αj

Cj(b+

|ξ|)α_|ˆ_s

μ(ξ)|dξ≤C. This completes the proof of Lemma ..

At the end of this section, we introduce two theorems which are important for our dis-cussions. As in [], we useSTAR(A) to denote all setsB⊆[, ]withC boundary∂B

given by

β(θ) = ρ(θ)cosθ

ρ(θ)sinθ

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cμ:=f,sμ,Mj:={(j,l,k,d),|k| ≤j+, –j≤l≤j– ,d= , }and

R(j,ε) =μ∈Mj:|cμ|>ε

.

Then withR(j,ε) standing for the cardinality ofR(j,ε), the following conclusion holds []. Theorem . For f ∈ε(A),R(j,ε)≤Cε– _and

μ∈Mj

|cμ|≤C–j.

Theorem .[] Let X∼N(u, )and t=log(η–₎_with_{ <}_η≤ .Then

ETs(X,t) –u



=logη–+ η+minu, ,

where N(u, )denotes the normal distribution with mean u and variance,while Ts(y,t) :=

sgn(y)(|y|–t)+is the soft thresholding function.

3 Main theorem

In this section, we give an approximation result, which extends the result [, Theorem .] from the Radon transform to a family of linear operators. To do that, we introduce a set N(ε) of signiﬁcant shearlet coeﬃcients as follows. Let

s=



 + α

log_ε–, s=



 + α

log_ε–,

andj=s,j=s. DeﬁneN(ε) :=M(ε)∪N(ε)⊆M, where N(ε) =μ=k∈Z:|k| ≤j+_;

M(ε) =μ= (j,l,k,d) :j≤j≤j, –j≤l≤j– ,|k| ≤j+,d= , 

.

Consider the modelY=Kf +εW with (K*_Kf₎∧₍_ξ_{) = (}_b₊_|_ξ_|₎–α_fˆ₍_ξ_{). Lemma . tells}

thatyμ:= αjY,Uμ=f,sμ+εαjnμ, wherenμis Gaussian noise with zero mean and

bounded varianceσ_μ=Uμ≤C[]. Letcμ=f,sμand˜f=

μ∈N(ε)c˜μsμwith

˜ cμ=

⎧ ⎨ ⎩

Ts(yμ,ε

log(N(ε))αj_σ

μ), μ∈N(ε);

, otherwise,

whereTs(y,t) is the soft thresholding function. Then the following result holds.

Theorem . Let f ∈ε(A) be the solution to Y =Kf +εW with (K*Kf)∧(ξ) = (b+

|ξ|)–α_fˆ₍_ξ₎_and_{f be deﬁned as above}˜ _._Then

sup

f∈ε₍_A₎

E˜f –f≤Clogε–ε

 

 +α ₍_ε→_).

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Proof Since{sμ,μ∈M}is a Parseval frame,f=

μ∈Mcμsμandf˜=

μ∈N(ε)c˜μsμ,˜f–f =

μ∈M(˜cμ–cμ)sμ. Moreover,˜f –f=

μ∈M|˜cμ–cμ|and

E˜f –f=

μ∈N(ε)

E| ˜cμ–cμ|+

μ∈N(ε)C

|cμ|. (.)

In order to estimate_μ_∈_N₍_ε₎C|cμ|, one observes

μ∈Mj|cμ|

_≤_C_–j_{due to Theorem ..}

Then_j_>_j_μ∈Mj|cμ|

_≤_C

j>j

–j_≤_C_–j_{. By }j_ε–    +α_,

j>j

μ∈Mj

|cμ|ε

 

 +α_. _(.)

(Here and in what follows,ABdenotesA≤CBfor some constantC> ).

Next, one considerscμforj≤j≤jand|k| ≥j+. Note that|ψ(d)(x)| ≤Cm( +|x|)–m

(d= , ,m= , , . . .). Then|ψ_j(_,d_l_,)_k(x)| ≤Cm

 j( +|Bl

dA j

dx–k|)–m. Sincef ∈ε(A),suppf⊂

Q:= [, ]and

f,ψ_j(_,d_l_,)_k≤Cm

 j_f_∞

Q

 +Bl dA

j dx–k

–m

dx.

On the other hand,|Bl dA

j

dx| ≤ BldA j

d|x| ≤j|x| ≤

√

j_for_x_∈_Q

. Hence, ( +|BldA j dx–

k|)–m≤( +|k|–|Bl_dAj_dx|)–m≤(|k|–√j)–mfor|k| ≥j+. Moreover,_|_k_|≥_j+|cμ|≤

j

|k|≥j+(|k|– √

j₎–m_{= }j∞ n=

j+n_≤|_k_|≤_j+n+–mj(n–

√

)–m _j_–mj_×

∞

n=(j+n+)(n– √

)–m _j_–j(m–)_{, since} _m _{can be chosen big enough.}

There-fore,

j

j=j

j_–

l=–j

|k|≥j+

|cμ|≤Cm

∞

j=j

j–mj–j≤_–j_ε  

 +α _(.)

due to the choice ofj. The similar (even simpler) arguments show_|k|≥j+|f,ϕj,k|

ε

 

 +α _with_ϕ_j

,k(x) = 

j_ϕ_(j_x_–_k_{). This with (.) and (.) leads to}

μ∈N(ε)C

|cμ|ε

 

 +α. (.)

Finally, one estimates μ∈N(ε)E| ˜cμ–cμ|. By the deﬁnition of yμ, ε––αjσμ–yμ∼ N(ε––αj_σ–

μ cμ, ). Applying Theorem . withη–=N(ε), one obtains that

ETs

ε––αjσ_μ–yμ,

logN(ε)–ε––αjσ_μ–cμ



=logN(ε)+  

N(ε)+min

ε––αj_σ–

μ c



μ, 

(9)

Hence, E|Ts[yμ,εαjσμ

log(N(ε))] – cμ| [log(N(ε)) + ][ε 

αj_σ

μ

N(ε) + min{cμ,

εαj_σ

μ}]. Byc˜μ=Ts[yμ,εαjσμ

log(N(ε))] forμ∈N(ε), one knows that

E

μ∈N(ε)

| ˜cμ–cμ|

logN(ε)+ 

×

ε

μ∈N(ε)

αj_σ

μ

N(ε)+

μ∈N(ε)

mincμ,ε αj_σ

μ

. (.)

Note that N(ε)∩Mj ⊂ {(j,l,k,d) : |k| ≤j+,|l| ≤j}. Then N(ε)≤ Cj≤j

j

j_ε

–_

 +α, andlog(N(_ε))   +α

log(ε–)log(ε–). Since{σμ:μ∈M}is uniformly

bounded, [log(N(ε)) + ]ε_μ_∈_N₍_ε₎

αj_σ

μ

N(ε) log(ε

–₎_ε_αj_{. This with the choice of }j shows that

logN(ε)+ ε

μ∈N(ε)

αj_σ

μ

N(ε)log

ε–ε

 

 +α log_ε–_ε  

 +α. (.)

It remains to estimate [log(N(ε)) + ]μ∈N(ε)min{cμ,εαjσμ}. Clearly,

μ∈N(ε)

mincμ,ε

αj₌

{μ∈N(ε):|cμ|≥αjε}

αjε+ {μ∈N(ε):|cμ|<αjε}

|cμ|. (.)

By Theorem .,_{μ∈Mj:|cμ|≥αjε}

αj_ε_(αj_ε₎–__αj_ε_αj_ε_{. Hence,}

{μ∈N(ε):|cμ|≥αjε}

αjε=

j

j=j

{μ∈Mj:|cμ|≥αjε}

αjεαj_ε_. _(.)

On the other hand,_{_μ_∈_N₍_ε_):_|_c_μ_|_<αj_ε_}|cμ|=

j

j=j _∞

n=

{αj–n–_ε_<_|_c

μ|≤αj–nε}|cμ|.

Ac-cording to Theorem .,R(j, αj–n–_ε₎_–_(αj–n–)_ε–_ _and

{αj–n–_ε_<_|_c

μ|≤αj–nε}

|cμ|–



(αj–n–)_ε–_(αj–n)_ε_αj_–n__ε_.

Therefore,

{μ∈N(ε):|cμ|<αj_ε_}

|cμ|≤

j

j=j ∞

n=

αj_–n_ε≤_αj_ε_.

Combining this with (.) and (.), one knows thatμ∈N(ε)min{cμ,εαj}

 αj_ε_. Furthermore,

μ∈N(ε)

mincμ,ε αj_ε

 α+ _

(10)

thanks to j_ε–  α+ _

. Now, it follows from (.), (.) and (.) that

μ∈N(ε)

E| ˜cμ–cμ|log

ε–ε

 α+ _

.

This with (.) and (.) leads to the desired conclusionsup_f_∈_ε₍_A₎E˜f–f≤Clog(ε–)×

ε

 

 +α. The proof is completed.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

LH and YL ﬁnished this work together. Two authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Applied Mathematics, Beijing University of Technology, Beijing, 100124, P.R. China.}2_{Department of Basic}

Courses, Beijing Union University, Beijing, 100101, P.R. China.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 11271038) and the Natural Science Foundation of Beijing (No. 1082003).

Received: 14 August 2012 Accepted: 18 December 2012 Published: 7 January 2013 References

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doi:10.1186/1029-242X-2013-11